Expokit is a software package designed to compute matrix exponentials, particularly for small matrices and large sparse matrices. It provides routines for computing the matrix exponential of a small matrix in full, the action of a large sparse matrix exponential on a vector, and the solution of systems of linear ODEs with constant inhomogeneity. The package handles real and complex matrices, including symmetric and Hermitian matrices, and is particularly useful for Markov chains. The core of the sparse routines is based on matrix-free Krylov subspace projection methods (Arnoldi and Lanczos processes), which enable efficient handling of large-dimensional sparse matrices. The software also addresses the computation of transient states in Markov chains, which are crucial for understanding the behavior of many physical systems. Expokit includes routines for both homogeneous and nonhomogeneous cases, and it provides error control mechanisms to ensure accuracy. The package is available through a World Wide Web site and includes MATLAB counterparts for experimental purposes.Expokit is a software package designed to compute matrix exponentials, particularly for small matrices and large sparse matrices. It provides routines for computing the matrix exponential of a small matrix in full, the action of a large sparse matrix exponential on a vector, and the solution of systems of linear ODEs with constant inhomogeneity. The package handles real and complex matrices, including symmetric and Hermitian matrices, and is particularly useful for Markov chains. The core of the sparse routines is based on matrix-free Krylov subspace projection methods (Arnoldi and Lanczos processes), which enable efficient handling of large-dimensional sparse matrices. The software also addresses the computation of transient states in Markov chains, which are crucial for understanding the behavior of many physical systems. Expokit includes routines for both homogeneous and nonhomogeneous cases, and it provides error control mechanisms to ensure accuracy. The package is available through a World Wide Web site and includes MATLAB counterparts for experimental purposes.